Re: error correcting/detecting code question

Bart Vandewoestyne wrote:
I have course notes on error detecting and correcting codes that
state the following two well known facts:

Fact 1:
"A block-code can detect s errors if and only if d >= s+1"

Fact 2:
"A block-code can correct t errors if and only if d >= 2t+1"

where d is the distance of the code.

Now a little bit further in these course notes, it is written:

"In general, a code can only detect errors or only correct errors
or a combination of both. The requirement for a code to be able
to detect s errors and at the same time correct t (<=s) errors is
that d>=s+t+1

Example:
Suppose we have a code with d=5. This code can...
... detect 4 errors and correct 0 errors.
OR ... detect 3 errors and correct 1 error.
OR ... detect 2 errors and correct 2 errors."


But I am puzzled by this, because when d=5 one could argue that
by Fact 1 we have that 4 errors can be detected and at the same
time one can argue by Fact 2 that 2 errors can be corrected. So
this would mean the code can detect 4 errors and correct 2
errors, which is in contradiction with the above.

If there are two errors, you can correct them both. If there are 4 errors, you don't get enough information to correct any of them.

What am I misinterpreting here? Which interpretation is the
right one? Do there exist codes that can detect 4 errors and
correct none, or is it possible with d=5 to detect 4 errors and
to correct 2 errors?

The knowledge that there are indeed at most two errors has to come from outside the error correction system. When there are three or more errors, you can still perform the procedure that would correct two errors, but its result won't be the original message.

--Mike Amling
.

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